• Corpus ID: 244714544

Near critical scaling relations for planar Bernoulli percolation without differential inequalities

@inproceedings{DuminilCopin2021NearCS,
  title={Near critical scaling relations for planar Bernoulli percolation without differential inequalities},
  author={Hugo Duminil-Copin and Ioan Manolescu and Vincent Tassion},
  year={2021}
}
We provide a new proof of the near-critical scaling relation β = ξ1ν for Bernoulli percolation on the square lattice already proved by Kesten in 1987. We rely on a novel approach that does not invoke Russo’s formula, but rather relates differences in crossing probabilities at different scales. The argument is shorter and more robust than previous ones and is more likely to be adapted to other models. The same approach may be used to prove the other scaling relations appearing in Kesten’s work. 

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References

SHOWING 1-10 OF 14 REFERENCES
Scaling relations for 2D-percolation
We prove that the relations 2D-percolation hold for the usual critical exponents for 2D-percolation, provided the exponents δ andv exist. Even without the last assumption various relations
CRITICAL EXPONENTS FOR TWO-DIMENSIONAL PERCOLATION
We show how to combine Kesten's scaling relations, the determination of critical exponents associated to the stochastic Loewner evolution process by Lawler, Schramm, and Werner, and Smirnov's proof
Near-critical percolation in two dimensions
We give a self-contained and detailed presentation of Kesten's results that allow to relate critical and near-critical percolation on the triangular lattice. They constitute an important step in the
The scaling limits of near-critical and dynamical percolation
We prove that near-critical percolation and dynamical percolation on the triangular lattice $\eta \mathbb{T}$ have a scaling limit as the mesh $\eta \to 0$, in the "quad-crossing" space $\mathcal{H}$
A note on percolation
SummaryAn improvement of Harris' theorem on percolation is obtained; it implies relations between critical points of matching graphs of the type of the one stated by Essam and Sykes. As another
Two-Dimensional Critical Percolation: The Full Scaling Limit
We use SLE6 paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the
Planar random-cluster model: scaling relations
This paper studies the critical and near-critical regimes of the planar random-cluster model on $\mathbb Z^2$ with cluster-weight $q\in[1,4]$ using novel coupling techniques. More precisely, we
Percolation ?
572 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Percolation is a simple probabilistic model which exhibits a phase transition (as we explain below). The simplest version takes place on Z2, which we view
The critical probability of bond percolation on the square lattice equals 1/2
We prove the statement in the title of the paper.
...
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